Budan's work on approximation was studied by Horner in preparing his celebrated article in the Philosophical Transactions of the Royal Society of London in 1819 that gave rise to the term Horner's method; Horner comments there and elsewhere on Budan's results, at first being sceptical of the value of Budan's work, but later warming to it.
Chinese algebra discovered numerical evaluation (Horner's method, re-established by William George Horner in the 19th century) of arbitrary degree algebraic equation with real coefficients.
His contribution to approximation theory is honoured in the designation Horner's method, in particular respect of a paper in Philosophical Transactions of the Royal Society of London for 1819.
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He became a lecturer in mathematics at the University in Sydney, where he died after only a few years - he had been advised to try a change of climate on account of tuberculosis.
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